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| '''Alan M. Frieze''' (born 25 October 1945 in [[London, England]]) is a [[professor]] in the Department of Mathematical Sciences at [[Carnegie Mellon University]], [[Pittsburgh]], [[United States]]. He graduated from the [[University of Oxford]] in 1966, and obtained his PhD from the [[University of London]] in 1975. His research interests lie in [[combinatorics]], [[discrete optimization]] and [[theoretical computer science]]. Currently, he focuses on the probabilistic aspects of these areas; in particular, the study of the asymptotic properties of [[random graphs]], the average case analysis of algorithms, and [[randomized algorithms]]. His recent work has included [[approximate counting]] and volume computation via [[random walks]]; finding edge disjoint paths in [[expander graphs]], and exploring [[anti-Ramsey theory]] and the stability of [[routing algorithms]].
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| ==Key contributions==
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| Two key contributions made by Alan Frieze are:
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| (1) polynomial time algorithm for approximating the volume of [[convex bodies]]
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| (2) algorithmic version for [[Szemerédi regularity lemma]]
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| Both these algorithms will be described briefly here.
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| ===Polynomial time algorithm for approximating the volume of convex bodies===
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| The paper
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| <ref name=JACM91>
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| {{cite news
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| |author= M.Dyer, A.Frieze and R.Kannan
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| |title=A random polynomial-time algorithm for approximating the volume of convex bodies
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| |journal= Journal of the ACM
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| |volume = 38
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| |number =1
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| |pages =1–17
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| |year= 1991
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| |url=http://portal.acm.org/citation.cfm?id=102783}}
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| </ref>
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| is a joint work by [[Martin Dyer]], Alan Frieze and [[Ravindran Kannan]].
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| The main result of the paper is a randomized algorithm for finding an <math>\epsilon</math> approximation to the volume of a convex body <math>K</math> in <math>n</math>-dimensional Euclidean space by assume the existence of a membership oracle. The algorithm takes time bounded by a polynomial in <math>n</math>, the dimension of <math>K</math> and <math>1/\epsilon</math>.
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| The algorithm is a sophisticated usage of the so-called [[Markov Chain Monte Carlo]] (MCMC) method.
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| The basic scheme of the algorithm is a nearly uniform sampling from within <math>K</math> by placing a grid consisting ''n''-dimensional cubes and doing a random walk over these cubes. By using the theory of
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| [[Markov chain mixing time|rapidly mixing Markov chains]], they show that it takes a polynomial time for the random walk to settle down to being a nearly uniform distribution.
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| ===Algorithmic version for Szemerédi regularity partition===
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| This paper
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| <ref>
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| {{cite news
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| |author= A.Frieze and R.Kannan
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| |title=A Simple Algorithm for Constructing Szemere'di's Regularity Partition
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| |journal= Electr. J. Comb.
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| |volume = 6
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| |year= 1999
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| |url=http://www.math.cmu.edu/~af1p/Texfiles/svreg.pdf}}
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| </ref>
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| is a combined work by Alan Frieze and [[Ravindran Kannan]]. They use two lemmas to derive the algorithmic version of the [[Szemerédi regularity lemma]] to find an <math>\epsilon</math>-regular partition.
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| <br />'''Lemma 1:''' <br />Fix k and <math>\gamma</math> and let <math>G=(V,E)</math> be a graph with <math>n</math> vertices. Let <math>P</math> be an equitable partition of <math>V</math> in classes <math>V_0, V_1, \ldots ,V_k</math>. Assume <math>|V_1| > 4^{2k}</math> and <math>4^k >600 \gamma ^2</math>. Given proofs that more than <math>\gamma k^2</math> pairs <math>(V_r,V_s)</math> are not <math>\gamma</math>-regular, it is possible to find in O(n) time an equitable partition <math>P'</math> (which is a refinement of <math>P</math>) into <math>1+k4^k</math> classes, with an exceptional class of cardinality at most <math>|V_0|+n/4^k</math> and such that <math>\operatorname{ind}(P')\geq \operatorname{ind}(P) + \gamma^5/20</math>
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| <br />'''Lemma 2:''' <br />Let <math>W</math> be a <math>R \times C</math> matrix with <math>|R|=p</math>, <math>|C|=q</math> and <math>\|W\|_\inf\leq1</math> and <math>\gamma</math> be a positive real.
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| <br />(a) If there exist <math>S \subseteq R</math>, <math>T \subseteq C</math> such that <math>|S|\geq\gamma p</math>, <math>|T|\geq\gamma q</math> and <math>|W(S,T)|\geq\gamma |S||T|</math> then <math>\sigma_1(W)\geq\gamma^3\sqrt{pq}</math>
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| <br />(b) If <math>\sigma_1(W)\geq\gamma\sqrt{pq}</math>, then there exist <math>S\subseteq R</math>, <math>T\subseteq C</math> such that <math>|S|\geq\gamma'p</math>, <math>|T|\geq\gamma'q</math> and <math>W(S,T)\geq\gamma'|S||T|</math> where <math>\gamma'=\gamma^3/108</math>. Furthermore <math>S</math>, <math>T</math> can be constructed in polynomial time.
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| <br />These two lemmas are combined in the following algorithmic construction of the [[Szemerédi regularity lemma]].
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| <br />'''[Step 1]''' Arbitrarily divide the vertices of <math>G</math> into an equitable partition <math>P_1</math> with classes <math>V_0,V_1,\ldots,V_b</math> where <math>|V_i|\lfloor n/b \rfloor</math> and hence <math>|V_0|<b</math>. denote <math>k_1=b</math>.
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| <br />'''[Step 2]''' For every pair <math>(V_r,V_s)</math> of <math>P_i</math>, compute <math>\sigma_1(W_{r,s})</math>. If the pair <math>(V_r,V_s)</math> are not <math>\epsilon-</math>regular then by Lemma 2 we obtain a proof that they are not <math>\gamma=\epsilon^9/108-</math>regular. | |
| <br />'''[Step 3]''' If there are at most <math>\epsilon | |
| \left(
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| \begin{array}{c}
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| k_1\\
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| 2 \\
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| \end{array}
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| \right)</math> pairs that produce proofs of non <math>\gamma-</math>regularity that halt. <math>P_i</math> is <math>\epsilon-</math>regular.
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| <br />'''[Step 4]''' Apply Lemma 1 where <math>P=P_i</math>, <math>k=k_i</math>, <math>\gamma=\epsilon^9/108</math> and obtain <math>P'</math> with <math>1+k_i4^{k_i}</math> classes
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| <br />'''[Step 5]''' Let <math>k_i+1 = k_i4^{k_i}</math>, <math>P_i+1=P'</math>, <math>i=i+1</math> and go to Step 2.
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| ==Awards and honors==
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| In 1991, Frieze received the [[Fulkerson Prize]] in Discrete Mathematics (Jointly with Martin Dyer and Ravi Kannan for the paper "A random polynomial time algorithm for approximating the volume of convex bodies" in the Journal of the Association for Computing Machinery) awarded by the American Mathematical Society and the Mathematical Programming Society.
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| In 1997 he was a Guggenheim Fellow
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| In 2000, he received the IBM Faculty Partnership Award
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| In 2006 he jointly received (with [[Michael Krivelevich]]) the Professor Pazy Memorial Research Award from the United States-Israel Binational Science Foundation.
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| In 2011 he was selected as a SIAM Fellow.
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| In 2012 he was selected as an AMS fellow.<ref>[http://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 2012-12-29.</ref>
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| ==References and external links==
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| <references/>
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| * [http://www.math.cmu.edu/~af1p/index.html Alan Frieze's web page]
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| * [http://www.math.cmu.edu/~af1p/Texfiles/oldvolume.pdf Fulkerson prize-winning paper]
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| * [http://www.cs.cmu.edu/~cfrieze/index.html Carol Frieze's web page]
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| * [http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/f/Frieze:Alan_M=.html Alan Frieze's publications at DBLP]
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| * [https://tspace.library.utoronto.ca/browse?type=author&order=ASC&rpp=20&value=Frieze%2C+A. Certain self-archived works are available here]
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME =Frieze, Alan
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = British mathematician
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| | DATE OF BIRTH =25 October 1945
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| | PLACE OF BIRTH =
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| | DATE OF DEATH =
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| | PLACE OF DEATH =
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| }}
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| {{DEFAULTSORT:Frieze, Alan}}
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| [[Category:English mathematicians]]
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| [[Category:Living people]]
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| [[Category:Carnegie Mellon University faculty]]
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| [[Category:Theoretical computer scientists]]
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| [[Category:Fellows of the American Mathematical Society]]
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| [[Category:1945 births]]
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| [[Category:Guggenheim Fellows]]
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The writer is named Josiah Sorrell but he will not like when persons use his whole identify. A person of the matters he loves most is to go to ballet but he will not have the time currently. Wisconsin has often been his house. He is at present a receptionist. His wife and he keep a website. You may possibly want to test it out: http://www.lallardaitana.com/planos/es/free/nike-free-3.0-mujer.php
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